P-groups

P-groups, or *p-groups*, are a specific type of group in the field of abstract algebra, particularly in the study of group theory. A group \( G \) is classified as a p-group if the order (the number of elements) of the group is a power of a prime number \( p \). Formally, this can be expressed as: \[ |G| = p^n \] for some non-negative integer \( n \).

Coclass, also known as co-class or co-classification, is a mathematical concept primarily used in the field of group theory, more specifically in the study of finite groups and their subgroups. It refers to a particular classification of groups based on shared properties of their Sylow subgroups.

The term "Extra Special Group" is not widely defined in common literature, organizations, or terminology as of my last knowledge update in October 2021. It could refer to a specific organization, initiative, or group focusing on unique or niche areas, but without additional context, it's challenging to provide an accurate description.

The Focal Subgroup Theorem is a concept in the area of algebraic topology and group theory, particularly relating to finite group actions and their relationships to fixed point sets in topological spaces. In more detail, the Focal Subgroup Theorem often pertains to the study of groups acting on topological spaces and examines the interaction between the group action and the topology of the space.

In group theory, which is a branch of abstract algebra, a **P-group** is a type of group that plays an important role in the study of finite groups. Specifically, a P-group is defined as a group in which the order (the number of elements) of every element is a power of a prime number \( p \).

A **powerful \( p \)-group** is a special type of \( p \)-group (a group where the order of every element is a power of a prime \( p \)) that satisfies certain conditions regarding its commutator structure.

A **regular p-group** is a specific type of finite group that is defined in the context of group theory, particularly in relation to \( p \)-groups. A **\( p \)-group** is a group where the order (the number of elements) of the group is a power of a prime number \( p \).

In the context of finite group theory, a "special group" typically refers to a type of group that has specific properties. One common usage is related to **special linear groups**. However, the term could also refer to **special groups in the context of group extensions** or other specific constructions in group theory.

The Sylow theorems are a set of results in group theory, a branch of abstract algebra. They provide important information about the subgroups of a finite group, particularly regarding the existence and properties of p-subgroups, where p is a prime number.